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Final Report


AF Contract 44620-67-C-0049

January 1, 1967 to December 31, 1973



The participants with their current positions are as follows:

K.L. Chung, Stanford University
John B. Walsh, University of British Columbia
Charles Lamb, University of British Columbia
Erhan Cinlar, Northwestern University
Peter Brockwell, La Trobe University (Australia)

The following persons completed their doctorates under the super-
vision of Chung during the period of the project, listed below with
their current positions:

Arthur Pittenger, University of Maryland
Robert Smythe, University of Washington
Gaston Giroux, University of Sherbrooke (Canada)
Michael Chamberlain, University of Santa Clara

Pending completion of his doctorate is Chris Nevison.

Short term consultants include the following persons: H. Kesten,
D. Stroock, D. Austin, N. Jain, S. Lloyd, J. L. Doob, C. Dellacherie,
D. Burkholder, A. Garsia.

Publications of the participants are listed below:

K. L. Chung

(1) (With John B. Walsh) "To reverse a Markov process," Acta Math.,
Vol. 123, pp. 225-251, 1969.
(2) Boundary Theory for Markov Chains, xvi+94 pages, Princeton
University Press, 1970.
(3) "On diverse questions of time reversing in Markov chains (and
processes)," Prod. of the Twelfth Biennial Seminar Canadian
Congress of Math., pp. 165-175.
(4) "Boundary behavior of Markov chains and its contributions to general
processes," Invited lecture at the International Congress of Mathe-
maticians, Nice, 1970; Actes du Congrbs, Vol. 2, pp. 499-506, 1971.






Final Report Chung
2



(5) "A simple proof of Doob's convergence theorem," Seminaire de
Probabilites V, Universite de Strasbourg, Springer-Verlag, 1971.
(6) "On the fundamental hypotheses of Hunt processes," Symposia
Mathematica, Istituto Nazionale di Alta Matematica, Vol. IX,
pp. 43-52, 1972.
(7) "Poisson process as renewal process," Period. Mat., Vol. 2,
pp. 41-48, 1972.
(8) "An expression for canonical entrance laws," (to appear).

(9) "Some universal field equations," S&minaire de Probabilites VI,
pp. 90-97, Universite de Strasbourg, Springer-Verlag, 1972.
(10) "Crudely stationary counting processes", Amer. Math. Monthly,
vol. 79, pp. 867-877, 1972.
(11) "Probabilistic approach to the equilibrium problem in potential
theory" (to appear in Ann. Inst. Fourier).
(12) (With Brockwell) "Emptiness times for a dam with stable imput
and general release function," (to appear).


John B. Walsh

(1) "The Martin boundary and completion of Markov chains," Z. Wahr-
scheinlichkeitstheorie und Verw. Gebiete, Vol. 14, pp. 169-199, 1970.
(2) "Some remarks on the Feller property," Ann. Math. Statist., Vol. 41,
pp. 1672-1683, 1970.
(3) "Time reversal and the completion of Markov processes," Invent.
Math., Vol. 10, pp. 57-81, 1970.


Charles W. Lamb

(1) "On the construction of certain transition functions," Ann. Math.
Statist., Vol. 42, pp. 439-450, 1971.
(2) "Decomposition and construction of Markov chains," Z. Wahrschein-
lichkeitstheorie und Verw. Gebiete, Vol. 19, pp. 213-224, 1971.
(3) "A note on harmonic functions and martingales," Ann. Math. Statist.,
Vol. 42, pp. 2044-2049, 1971.


Erhan Cinlar

(1) "Theory of continuous storage with Markov additive inputs and a
general release rule," J. Math. Anal. Appl., Vol. 43, pp. 207-231, 1973.
(2) (With Jagers) "Two mean values which characterize the Poisson process,"
J. Appl. Probaility, Vol. 10, pp. 678-681, 1973.







Final Report Chung
3



Peter Brockwell

(1) "On the spectrum of a class of matrices arising in storage theory,"
Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Vol. 25, pp. 253-260,
1973.
(2) "Deviations from monotonicity of a Wiener process with drift,"
(to appear in J. Appl. Probability).


Detailed descriptions of work done have been included in the annual
status reports. To summarize, the major achievements are:

(I) Boundary theory for Markov chains, see under Chung (2) and (4);

(II) Time reversing of Markov processes, see under Chung (1); and
Walsh (2);

(III) Application of last exit distribution to equilibrium problem,
see under Chung (11).

(IV) Various applications to Poisson and point processes, see under
Chung (7), (10) and (12).

It was further planned to apply the techniques in treating boundaries,
time reversals and last exits to practical problems such as turbulence,
electromagnetic equilibrium, dam and other storage problems. Initial
progress was made in Chung and Brockwell (see under Chung (12)). Termi-
nation of the project has cut it short.




Professor Kai Lai Chung
Principal Investigator





Unclassified
Security Classification
DOCUMENT CONTROL DATA R & D
(Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified)
1. ORIGINATING ACTIVITY (Corporate author) l2a. REPORT SECURITY CLASSIFICATION
Mathematics Department Unclassified
Stanford University 2b. GROUP
Stanford, CA 94305
3. REPORT TITLE
RESEARCH IN PROBABILISTIC TECHNIQUES FOR SYSTEMS ANALYSIS (MARKOV PROCESSES)


4. DESCRIPTIVE NOTES (Type of report and Inclusive dates)
Scientific Final
S. AUTHOR(S) (First name, middle Initial, last name)


Kai Lai Chung

S. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS
February 1974 3 22
8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBERS)
F 44620-67-C-0049
b. PROJECT NO.
9769
C. 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned
this report)
d.
10. DISTRIBUTION STATEMENT



1I. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
Air Force Office of Scientific Research
Tech, Other 1400 Wilson Blvd.
Arlington, VA 22209
13. ABSTRACT


Major results are obtained for the boundary theory of Markov chains,

time reversing of Markov processes, and application of last exit

distribution to equilibrium problems. Practical applications are

envisaged for Poisson and point processes; turbulence, electromagnetic

and storage problems.











al \ae .


DD .."" 1441723


Unclassified
Security Classlficapfon









October 8, 1974


Dr. William G. Rosen
Program Director for
Modern Analysis and Probability
National Science Foundation
Washington, D.C. 20550

Dear Dr. Rosen:

The following is a summary of work performed to date on NSF grant
41710X, as well as a request for continued funding for 1975-76. A budget
for the period March 1, 1975 to Februairy 28, 1976 is attached. Owing to
the circumstance that Dr. Charles Lamb was unable to participate, it is
estimated that there will be an unspent balance of approximately $3,000
in the current funds.
During the summer of 1974, Dr. John B. Walsh and I worked together
for two months. We finished a paper cont dining a new and simpler proof
of P. A. Meyer's celebrated theorem that a stopping time is predictable
where the sample path is continuous in a Hunt process. As a by-product
we found a single family of stopping times which announce a predictable
time simultaneously for all probability laws. Our next area of investi-
gation is the relation between equilibrium, energy and duality. I found
that if the potential equilibrium density is symmetric, then the equi-
librium measure obtained by the last exit method (Chung, 1972) in fact
has the minimizing properties as in the classical cases of Gauss and
Frostman. The asymmetric case seems much harder but a similar variational
approach has been suggested by discrete analogues in Markov chains. I was
invited to give a talk on the subject by the London Mathematical Society
at the Durham Symposium on Functional Analysis and Stochastic Processes,
where I learned of some new possibilities of approaching .ihe problem.
Work will be continued in this direction. Dr. Walsh did very substantial
work on duality theory by introducing a new method based on Doob's h-path
process but coupled with the idea of last exits near the lifetime. He
has completed a paper entitled, "The cofine topology revisited" which will
be published. We intend to persue various applications of this method.
Dr. Walsh will be in Europe next summer, but can come for short
periods. We propose to have Dr. Pierre van Moerbeke participate in the
project for two months in the summer of 1975. He will be visiting
associate professor in the mathematics department here and fits in very
well with the program. He studied with Kac and McKean and will bring
his knowledge of diffusion processes and differential equations to bear
on the analytic applications of modern probability theory. He will also
work on optimal boundary problems. His vitae and list of publications
are enclosed herewith for your evaluation and approval.
I plan to take a sabbatical leave in the academic year of 1975-76.
Stanford University pays half of my salary for the period. I am









Dr. William G. Rosen


requesting your support for one sixth of my academic salary which will
allow me to have two quarters of leave. The change in my academic
salary reflects an annual increase of approximately 5% effective
September 1, 1974. A similar increase can be projected for September 1,
1975. During my leave I plan to visit several universities in the east
and abroad. I have applied for a Guggenheim fellowship to make this
possible. There is no other pending application for other support.
If you wish any further information regarding this report and
renewal request, please let me know.
Thanking you for your support, I am,

Sincerely yours,



Kai Lai Chung
Professor of Mathematics

KIIC:ca
Atts.


October 8, 1974












Three reprints each of ##3, 4, 7 above are hereby enclosed, as well as a

copy of #9. Reprints of ##l and 2 were sent to you previously. Reprints ##5

and 11 should be arriving soon and will be sent.

Summaries of previous work were given in my annual reports for 1974 and

1975. The work done by Dr. Raq and myself during 1976 was reported in the

new proposal submitted to you last October. Dr. Rao obtained a nice proof of

Kanda's noted result about polar sets for Levy processes, using the energy

principle. My work on a modified energy concept for probabilistic potential

theory is continuing. The work on certain Wiener functionals associated with

the Schridinger equations, initiated in the one-dimensional case in #9 above,

is now being extended to the much more difficult case of high dimensions by

Dr. Varadhan and myself. The work on Brownian excursions and related topics

(##2, 4, 5, 7) is further developing. Lecture notes on them are being prepared

and expanded by Dr. Balkema and myself for publication in monograph form.


K, L. Chung, Principal Investigator






STANFORD UNIVERSITY
STANFORD, CALIFORNIA 94305

DEPARTMENT OF MATHEMATICS
October 14, 1975


Dr. William G. Rosen
Program Director for Modern Analysis
and Probability
National Science Foundation
Washington, D.C. 20550

Reference: NSF Grant No. MPS74-00405 A01

Dear Dr. Rosen:

This is a summary of work performed to date on NSF grant MPS74-00405 A01,
and a request for continued funding for 1976-1977. A budget for the period
March 1, 1976- February 28, 1977 is attached.
The following papers have been published, accepted and submitted for
publication:
1. K. L. Chung and John B. Walsh: "Meyer's theorem on predictability",
Z. Wahrscheinlichkeitstheorie 29 (1974), 253-256.
2. K. L. Chung: "Maxima in Brownian excursions", Bull. Amer. Math.
Soc. 81 (1975), 742-745.
3. K. L. Chung: "Remarks on equilibrium potential and energy", Ann.
Inst. Fourier (in print).
4. K. L. Chung: "A bivariate distribution in regeneration", J. Appl.
Probability (in print).
5. K. L. Chung: "Excursions in Brownian motion" (submitted for
publication).
6. K. L. Chung and R. Durrett: "Upcrossing and local time" (to appear).
7. John B. Walsh: "The cofine topology revisited" (to appear).
8. Pierre van Moerbeke: "The spectrum of Jacobi matrices" (to appear).
9. Michael Steele: "Combinatorial entropy and uniform limit laws",
Doctoral Dissertation, Stanford University, 1975.

Reprints and preprints of the works listed above, except 9, are being
sent to you under separate cover. Mr. Steele received some support from the
project, obtained his doctorate and is now at the University of British Columbia.
The following manuscripts are being prepared:
10. K. L. Chung: "On the condenser problem"
11. K. L. Chung and P. van Moerbeke: "Brownian functional related to a
classical differential equation".

For the summer of 1976, support for Professor Murali Rao and Mr. Richard
Durrett is requested. At a recent conference I learned that Rao is working
on problems concerning the application of probability methods to potential







Dr. W. G. Rosen, NSF, Washington


theory, very much along the lines of my project. After seeing my previous
results on equilibrium potential, he is eager to participate in further
investigations. He is writing a book on Brownian motion and potentials,
and his knowledge and experience of the field will be most valuable. Mr.
Durrett is a student in the Department of Operations Research at Stanford
and has attended several of my courses and seminars. He is the best student
in probability that we have had in a number of years. He has already co-
operated with me on a finished paper (# 6 above) and has started on another
(# 10 above). He has essentially completed his doctoral dissertation (under
the supervision of Professor Iglehart) and would like to continue the work
on my project before taking an academic position.
Vitae for Rao and Durrett are herewith submitted for your approval.
If you wish any further information regarding this report and renewal
request, please let me know.
Thanking you for your support, I am,

Sincerely yours,



Kai Lai Chung
Professor of Mathematics

KLC:ca
Encls.


10-14-75









Report of Progress for Summer 1976


As mentioned in my earlier proposal for Summer 76, the subject of

probabilistic potential theory has received renewed interest. This was

the reason for initiating a research project on Classical-cum-Modern Potential

Theory which will culminate in a book that will delineate the classical material

which has had such intensive ramifications. The grant has helped me develop

a major position of the project. The exploration of related questions has

led to extension of known results concerning the maximum principle and the

representation of potentials.

In another direction Professor Chung and myself have been working

on the implications of the energy principle. The ideas we have developed

will lead to a generalization and simplification of a recent major result

of M. Kanda, namely that for some Levy processes semipolar sets are polar.







STANFORD UNIVERSITY
STANFORD. CALIFORNIA 94305

DEPARTMENT OF MATHEMATICS

October 18, 1977


Dr. William G. Rosen
National Science Foundation
Washington, D.C. 20550

Dear Dr. Rosen:

The following is a summary of the research done by Dr. Murali Rao and
myself during the past summer on NSF grant MCS77-01319.

(1) The notion of energy related to a potential density kernel u(x,y) is
extended from the symmetric case to the general case by a modification as follows.
For a smooth compact K with its equilibrium measure v, as previously established
in [1], the last-exit kernel L(x,dy) = u(x,y)v(dy) considered on K has a
stationary probability measure r such that irL = w. Define

p(y) = f 7r(dx)u(x,y) > 0

and put
-l
u (x,y) = u(x,y) (y)

When u is symmetric, 'p reduces to a constant, which may be taken to be one.
In the general case v minimizes the energy

I (X) = ffX(dx)u (x,y)X(dy)

among all signed measures X of finite variation, which are supported by K and
correspond to continuous additive functionals. Further consequences of the
modification are being studied in order to put energy concepts for nonsymmetric
kernels on a similar footing with the classical theory in the symmetric case.

(2) In an earlier paper [2], Rao has given the Riesz representation of
excessive functions in the setting of [1]. Under the additional assumption
(a) u(x,y) is bounded when x and y vary over disjoint compact sets;
he now proves the uniqueness of the representation. From this he deduces that
Hunt's Hypothesis (B) is satisfied. Finally, under the further assumption
(b) there exists a strictly positive continuous function such that
y f m(dx)f(x)u(x,y) belongs to Co;
he proves that there is a dual which is a Feller process. Thus we now have a
class of Markov processes in duality under easily verifiable conditions.
I have simplified some of the proofs and compared the results with those
obtained by John B. Walsh in a quite different setting. For instance, in the
latter approach u(.,y) is assumed to be minimal excessive for each y,
whereas this fact can be deduced from our assumptions. These very recent
results will be written up for publication after further consolidation.






Dr. W. G. Rosen


(3) Jointly with Dr. Varadhan, I have completed some work on diffusion
and the Schrbdinger equation. This utilizes the techniques of stochastic
integrals and improves upon previous results by Dr. van Moerbeke and myself.
A preprint of the paper is enclosed. The extension to high dimensions has
already begun with the cooperation of Varadhan.

(4) A paper by Rao, entitled "On a result of M. Kanda", which was started
in the summer of 1976 on our project, has been accepted for publication in the
Zeitschrift fUr Wahrscheinlichkeitstheorie.

Thanking you for your support,

Sincerely yours,



Kai Lai Chung














[1] K. L. Chung, Probabilistic approach in potential theory to the equilibrium
problem, Ann. Inst. Fourier 23 (1973), 313-322.

[2] Murali Rao, Excessive functions as potentials of measures, J. London
Math. Soc. (to appear).


10-18-77







STANFORD UNIVERSITY
STANFORD, CALIFORNIA 94305

DEPARTMENT OF MATHEMATICS

September 26, 1977


Dr. William G. Rosen
National Science Foundation
Washington, D.C. 20550

Dear Dr. Rosen:

The following is a summary of the research done by Dr. Murali Rao and
myself during the past summer on NSF grant MCS77-01319.

(1) The notion of energy related to a potential density kernel u(x,y) is
extended from the symmetric case to the general case by a modification as follows.
For a smooth compact K with its equilibrium measure v, as previously established
in [1], the last-exit kernel L(x,dy) = u(x,y)v(dy) considered on K has a
stationary probability measure w such that 7rL = f. Define

p?(y) = f r(dx)u(x,y) > 0

and put
-1
u (x,y) = u(x,y)p(y) -

When u is symmetric, ( reduces to a constant, which may be taken to be one.
In the general case P minimizes the energy

I (X) = ffX(dx)up(x,y)X(dy)

among all signed measures X of finite variation, which are supported by K and
correspond to continuous additive functionals. Further consequences of the
modification are being studied in order to put energy concepts for nonsymmetric
kernels on a similar footing with the classical theory in the symmetric case.

(2) In an earlier paper [2], Rao has given the Riesz representation of
excessive functions in the setting of [1]. Under the additional assumption
(a) u(x,y) is bounded when x and y vary over disjoint compact sets;
he now proves the uniqueness of the representation. From this he deduces that
Hunt's Hypothesis (B) is satisfied. Finally, under the further assumption
(b) there exists a strictly positive continuous function such that
y -> f m(dx)f(x)u(x,y) belongs to Co;
he proves that there is a dual which is a Feller process. Thus we now have a
class of Markov processes in duality under easily verifiable conditions.
I have simplified some of the proofs and compared the results with those
obtained by John B. Walsh in a quite different setting. For instance, in the
latter approach u(.,y) is assumed to be minimal excessive for each y,
whereas this fact can be deduced from our assumptions. These very recent
results will be written up for publication after further consolidation.







Dr. W. G. Rosen


(3) Jointly with Dr. Varadhan, I have completed some work on diffusion
and the Schridinger equation. This utilizes the techniques of stochastic
integrals and improves upon previous results by Dr. van Moerbeke and myself.
A preprint of the paper is enclosed. The extension to high dimensions has
already begun with the cooperation of Varadhan.

(4) A paper by Rao, entitled "On a result of M. Kaida", which was started
in the summer of 1976 on our project, has been accepted for publication in the
Zeitschrift fUr Wahrscheinlichkeitstheorie.

Thanking you for your support,

Sincerely yours,



Kai Lai Chung














[1] K. L. Chung, Probabilistic approach in potential theory to the equilibrium
problem, Ann. Inst. Fourier 23 (1973), 313-322.

[2] Murali Rao, Excessive functions as potentials of measures, J. London
Math. Soc. (to appear).


9-26-77








Final Report


This is the final report on NSF grant MCS 74-00405-A02, March 1, 1976 to

February 28, 1977. The participants are:

K. L. Chung, Stanford University
John B. Walsh, now at University of British Columbia
Pierre van Moerbeke, now at University of Orsay
Murali Rao, University of Aarhus

The grant also supported in part the following doctorate students:

Chris Nevison, now at Colgate University
Mike Steele, now at University of British Columbia
Richard Durrett, now at University of California in Los Angeles

The following research papers were produced:

1. K. L. Chung and John B. Walsh: "Meyer's theorem on predictability",
Z. Wahrscheinlichkeitstheorie 29 (1974), 253-256.
2. K. L. Chung: "Maxima in Brownian excursions", Bull. Amer. Math. Soc. 81
(1975), 742-745.
3. K. L. Chung: "Remarks on equilibrium potential and energy", Ann. Inst.
Fourier 25 (1975), 131-138.
4. K. L. Chung: "A bivariate distribution in regeneration", J. Appl.
Probability 12 (1975), 837-839.
5. K. L. Chung: "Excursions in Brownian motion", Arkiv fUr Matematik 14
(1976), 155-177.
6. K. L. Chung and R. K. Getoor: "The condenser problem", to appear in
Ann. of Probability.
7. K. L. Chung and R. Durrett: "Downcrossings and local time", Z. Wahr-
scheinlichkeitstheorie 35 (1976), 147-150.
8. K. L. Chung: "A proof of Skorohod's lemma", Teor. Verojatnost. i Mat.
Statist. 15 (1976), 151-152.
9. K. L. Chung and Pierre van Moerbeke: "On certain Wiener functionals
associated with the Schrddinger equation", submitted for
publication.
10. John B. Walsh: "The cofine topology revisited", to appear.
11. Pierre van Moerbeke: "The spectrum of Jacobi matrices", Invent. Math. 37
(1976), 45-81.
12. Murali Rao: "On a result of M. Kanda", to appear.











Three reprints each of ##3, 4, 7 above are hereby enclosed, as well as a

copy of #9. Reprints of ##1 and 2 were sent to you previously. Reprints ##5

and 11 should be arriving soon and will be sent.

Summaries of previous work were given in my annual reports for 1974 and

1975. The work done by Dr. Rao and myself during 1976 was reported in the

new proposal submitted to you last October. Dr. Rao obtained a nice proof of

Kanda's noted result about polar sets for Levy processes, using the energy

principle. My work on a modified energy concept for probabilistic potential

theory is continuing. The work on certain Wiener functionals associated with

the Schridinger equations, initiated in the one-dimensional case in #9 above,

is now being extended to the much more difficult case of high dimensions by

Dr. Varadhan and myself. The work on Brownian excursions and related topics

(##2, 4, 5, 7) is further developing. Lecture notes on them are being prepared

and expanded by Dr. Balkema and myself for publication in monograph form.


K, L. Chung, Principal Investigator





STANFORD UNIVERSITY
STANFORD, CALIFORNIA 94305

DEPARTMENT OF MATHEMATICS September 11, 1978.


Dr. William G. Rosen
Program Director for Modern Analysis
and Probability
National Science Foundation
Washington, D.C. 20550

Reference: NSF Grant No. MCS77-01319 A01

Dear Dr. Rosen:

The following is a summary of the research work done during the past few
months on NSF grant MCS-7-01319 A01.

(1) Dr. K. Murali Rao and I made a thorough investigation of the class of
potential kernels introduced in [1]. We deal with a transient Hunt process with
a potential density function u satisfying the two conditions below:
(a) y -+ u(x,y)-1 is finite continuous for each x;
(b) u(x,y) = if and only if x = y.
We proved a number of results under these conditions. Some of these were known
previously only under strong duality and strong Feller assumptions. Our approach
has the advantage of tying simple verifiable analytic conditions to deep proba-
bilistic conclusions, thereby revealing structural relations sometimes "snowed
under" by other postulates. A basic result is the general existence of a version
w of the density u having the property that for each open set G:

w(x,y) = PGw(x,y) for all y in G

where PG is the balayage operator acting on x. Under (a) and (b) we showed
that there is a polar set Z such that w(.,y) = u(.,y) or = 0 according as
y Z or y E Z. This enables us to prove the uniqueness of the measure p
determining the potential Up, provided that p does not charge Z. A Riesz
decomposition is an easy consequence. Furthermore, we prove that Hunt's Hypothesis
(B) holds in our setting, namely: for any open G containing any compact K
we have
P K1 = PK1 ;

provided that u(-,y) is excessive for each y. This turns out to be a difficult
proposition. We are now encouraged to attack another one of Hunt's Hypothesis:
"Every semipolar set is polar", which goes back to Kellogg, and is a cornerstone
of the classical theory.

(2) Along the way we established the existence of a dual semigroup from
which a right continuous strong Markov dual process can be constructed. It is
strong Feller under a time change and satisfies the continuity principle. Yet
it does not have all the properties under the usual duality assumptions. We
intend to investigate the relation of this dual process to the sample-path
reverse which always exists according to the result by Chung and Walsh (1969),
and also the u-transform studied by Doob and Walsh (1976).






Dr. W. G. Rosen, NSF, Washington


(3) As a consequence of Hypothesis (B) mentioned above, it follows that
if v is the equilibrium measure on the compact K obtained by the method
of [1], the set function K -* v(K) induces a true Choquet capacity. Pre-
viously I proved this under the additional assumption that all paths are
continuous. Further development in this direction, in particular a theory
of energy as initiated in [4], depends on a better understanding of the
modified potential kernel introduced there.
The results summarized above partially supplant earlier ones reported
last October. Two preprints [2] and [3] have been sent to you in June and
a third one [4] is being sent under separate cover. These are being combined
and rewritten in a paper entitled "A new setting for potential theory", which
is under preparation. This will be presented by me in an invited talk in a
special session at the Claremont AMS meeting in October.
Mr. Joseph Glover was supported by the grant during the spring quarter
before he obtained his doctorate from UCSD La Jolla. (He is now lecturer
in the Department of Statistics at UC Berkeley.) Paper [5] is a joint
effort by him and me which was sent to you in June and has been submitted
for publication. In it a number of well-known results for the "right"
process are established for the "left" process. More sophisticated methods
are needed because several standard arguments are no longer valid in the
left case, but it came to us as a pleasant surprise that the famous theorems
by Cartan-Brelot-Doob, Hunt, Meyer and Dellacherie all hold true without
any sacrifices. As Meyer said, it certainly helps to be able to use the
left as well as the right hand. One good application has already been pointed
out by Rao, more should be forthcoming.
Thanking you for your support,

Sincerely,



Kai Lai Chung
Professor of Mathematics
KLC:ca


[1] K. L. Chung, Probabilistic approach in potential theory to the
equilibrium problem, Ann. Inst. Fourier 23 (1973), 313-322.
[2] K. L. Chung and Murali Rao, Potential theory without duality, Aarhus
University Preprint Series No. 22 (1977-8).
[3] K. L. Chung and Murali Rao, On existence of a dual process, ibid. No. 25 (1977-8)
[4] K. L. Chung and Murali Rao, Equilibrium and energy (to be sent).
[5] K. L. Chung and Joseph Glover, Left continuous moderate Markov processes
(submitted for publication).


9-11-78











Progress Report MCS-8301072


The following papers have been completed during the past year.


[1] K.L. Chung, The gauge and conditional gauge theorem. Seminaire de probabilites
XIX (1983/84), 496-503.

[2] K.L. Chung, Doubly-Feller process with multiplicative functional, Seminar on
Stochastic Processes, vol. 5, Birkhauser (to appear).

[3] M. Liao, Riesz representation and duality of Markov processes, Doctoral
Dissertation, Stanford University, August 1984.


In [1] it is proved that the conditional gauge theorem, which is an important

strengthening of the gauge theorem, actually follows from the latter provided an

estimate concerning a cordon sanitaire of small measure holds true. (See under

item (1) in the new research proposal.) Recently Chung gave a talk on this at the

Institute of Mathematics and its Applications at the University of Minnesota, pin-

pointing the problem. Afterwards, Cranston, Fabes, and Zhao were able to establish

the required estimate for a bounded Lipschitz domain. This great improvement over

previous results (first for C2, then for C11 domains) was made possible by the

reduction contained in [1].

In [2] it is proved that if a process has both the Feller and strong Feller

properties, then, if we attach a certain class of multiplicative functional to the

process and kill it off a regular open (not necessarily bounded!) set, the resulting

process has the same properties. In the special case of Brownian motion, this result

forms the base of analytic developments of the Feynman-Kac formula. The new result

does not require the continuity of paths and covers a wide class of multiplicative

functionals.

In [3] Liao generalized and simplified a number of previous results by Chung

and Rao. In particular, assumptions on the potential density function u(x,y) are

considerably reduced, and the existence of the exceptional set Z is thoroughly






2



investigated. In the duality case this is identified as the set of branching

points. Liao's thesis has many points of contact with current research in

Europe on axiomatic potential theory. He has returned to China and is now

on the faculty of Nankai University, Tianjin, after teaching for one semester

at the University of Florida.

Copies of [1] and [2] are being sent to your office. Reprint of [3] will

be sent when available.





Progress Report MCS-8301072


During the month of July, 1986 Chung worked with Thomas Salis-
bury on the extension of the gauge theorem to an arbitrary domain
in-Euclidean space with Martin boundary. It turned out that the
methods used by Chung in his paper "The Gauge and Conditional
Gauge Theorem" (Sgminaire des probabilit4s vol. XIX, 1983/84)
works equally well in the general case with appropriate modifica-
t. ions supplied by the construction of Martin boundary. See also
obeJ the separate progress report submitted by Salisbury. ,, i
Chung and Z. Zhao also proved the continuity of theAgauge function
up to the boundary. This and a number of related results will be
published in a monograph under preparation, tentatively entitled
"From Brownian Motion to Schrbdinger equation". The following note
deals with one of the problems originally proposed in the project:
K. L. Chung, Remark on the conditional gauge theorem
A preprint of this and a reprint of the following are sent to your
office under separate cover.
K. L. Chung, P. Lis and R. J. Williams, Comparison of probability
and classical methods for the Schridinger equation, Expo. Math
vol. 4 (1986), 271-278. -
This is a substantially revised version of an older manuscript.
Yr It deals with the eigenvalue (spectobn) connection of the gauge
b'theorem, miae a problem in the original proposal. Although hatxx
results Tor a larger class of functions q have been obtained (to
be publghed in the pe.o ,ee monograph mentioned above), the ana-
lytic connection is essentially the same.
A doctoral student, V. Papanicolaou is working on the mixed
boundary problem under the direction of Chung. Preliminary results
can be expected in the next progress report.


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